Solve (a^{2}+b^{2})^2+(a^2-b^2)^2

$$( a ^ { 2 } + b ^ { 2 } ) ^ { 2 } + ( a ^ { 2 } - b ^ { 2 } ) ^ { 2 }$$

Evaluate

$$2\left(a^{4}+b^{4}\right)$$

Solution Steps

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(a^{2}+b^{2}\right)^{2}.

$$\left(a^{2}\right)^{2}+2a^{2}b^{2}+\left(b^{2}\right)^{2}+\left(a^{2}-b^{2}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$a^{4}+2a^{2}b^{2}+\left(b^{2}\right)^{2}+\left(a^{2}-b^{2}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$a^{4}+2a^{2}b^{2}+b^{4}+\left(a^{2}-b^{2}\right)^{2}$$

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(a^{2}-b^{2}\right)^{2}.

$$a^{4}+2a^{2}b^{2}+b^{4}+\left(a^{2}\right)^{2}-2a^{2}b^{2}+\left(b^{2}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$a^{4}+2a^{2}b^{2}+b^{4}+a^{4}-2a^{2}b^{2}+\left(b^{2}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$a^{4}+2a^{2}b^{2}+b^{4}+a^{4}-2a^{2}b^{2}+b^{4}$$

Combine a^{4} and a^{4} to get 2a^{4}.

$$2a^{4}+2a^{2}b^{2}+b^{4}-2a^{2}b^{2}+b^{4}$$

Combine 2a^{2}b^{2} and -2a^{2}b^{2} to get 0.

$$2a^{4}+b^{4}+b^{4}$$

Combine b^{4} and b^{4} to get 2b^{4}.

$$2a^{4}+2b^{4}$$

Expand

$$2a^{4}+2b^{4}$$

Solution Steps

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(a^{2}+b^{2}\right)^{2}.

$$\left(a^{2}\right)^{2}+2a^{2}b^{2}+\left(b^{2}\right)^{2}+\left(a^{2}-b^{2}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$a^{4}+2a^{2}b^{2}+\left(b^{2}\right)^{2}+\left(a^{2}-b^{2}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$a^{4}+2a^{2}b^{2}+b^{4}+\left(a^{2}-b^{2}\right)^{2}$$

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(a^{2}-b^{2}\right)^{2}.

$$a^{4}+2a^{2}b^{2}+b^{4}+\left(a^{2}\right)^{2}-2a^{2}b^{2}+\left(b^{2}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$a^{4}+2a^{2}b^{2}+b^{4}+a^{4}-2a^{2}b^{2}+\left(b^{2}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$a^{4}+2a^{2}b^{2}+b^{4}+a^{4}-2a^{2}b^{2}+b^{4}$$

Combine a^{4} and a^{4} to get 2a^{4}.

$$2a^{4}+2a^{2}b^{2}+b^{4}-2a^{2}b^{2}+b^{4}$$

Combine 2a^{2}b^{2} and -2a^{2}b^{2} to get 0.

$$2a^{4}+b^{4}+b^{4}$$

Combine b^{4} and b^{4} to get 2b^{4}.

$$2a^{4}+2b^{4}$$